Macromolecules 2005, 38, 10571-10579
10571
Rheology of Polyethylenes with Novel Branching Topology Synthesized
by a Chain-Walking Catalyst
Rashmi Patil and Ralph H. Colby*
Department of Materials Science and Engineering, Pennsylvania State University,
University Park, Pennsylvania 16802
Daniel J. Read*
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT U.K.
Guanghui Chen and Zhibin Guan*
Department of Chemistry, University of California, Irvine, California 92697
Received June 29, 2005; Revised Manuscript Received October 4, 2005
ABSTRACT: Ethylene pressure was used to vary the molecular architecture of amorphous polyethylenes
synthesized with a palladium-bisimine catalyst. At low ethylene pressure, densely branched polymers
are formed, and their melt rheology indicates no entanglement even though the weight-average molar
mass is 370 000. Polymers produced at higher ethylene pressures have only slightly higher molar masses
but show entanglement effects in their rheology. NMR suggests similar levels of short-chain branching
(92-97 branches per 1000 carbons) in all the samples. A simple model of polymerization, based on the
proposed “chain-walking” mechanism for this catalyst, is used to generate structures via computer
simulation. While some of the experimental data are consistent with the predicted molecular structures,
there are discrepancies in (i) the scaling of radius of gyration with molecular weight and (ii) the variation
of terminal relaxation time with ethylene pressure, both of which suggest long-chain branching, which
is not predicted by the computer simulation.
Introduction
Traditional methods to produce dendrimers and hyperbranched polymers use condensation polymerization
of AB2 monomers where branching is introduced by the
structure of the monomer.1 Each addition of monomer
increases the number of active sites per chain. In most
cases, this conventional approach requires synthesis of
a specifically designed monomer. Free radical polymerizations of olefin monomers can have long-chain branching if the catalysts used encourage chain transfer,2 but
this results in broad molar mass distributions.
Without sacrificing the narrow molar mass distribution, single-site metallocene catalysts give excellent
control of the stereochemistry and molar mass of polyolefins.3 Hence, metallocene catalysis can provide a
useful tool for studying the influence of polymer architecture on rheological behavior. Brookhart and coworkers4-6 have demonstrated that catalysts involving
late transition metals, such as nickel and palladiumR-diimine catalysts, can lead to branched polyolefins via
a “chain-walking” mechanism of catalyst motion along
the growing polymer chain. By controlling the kinetic
competition between chain-walking and insertion, Guan
and co-workers demonstrated that polyethylenes with
a continuum of properties can be prepared from homopolymerization of ethylene and suggested that this
involved a range of branching topologies from linear to
hyperbranched to dendritic.7-9 The possibility of producing polymers with a continuum of topologies without
changing the chemical structure provides a unique
opportunity for fundamental studies of topological effects on polymer rheological properties. In this paper
we utilize the catalyst, illustrated in Figure 1, which
has been extensively studied for polymerization of
ethylene by Guan and co-workers.7-9
Figure 1. Ethylene coordination polymerization is catalyzed
by this Pd-R-diimine complex.
Accordingtotheproposed“chain-walking”mechanism,4-9
the catalyst remains attached to the growing chain for
the duration of the polymerization and grows the chain
via the following set of basic events: (i) ethylene
coordination and dissociation, in which an ethylene
molecule is trapped or released by the catalyst, (ii)
ethylene insertion, in which the trapped ethylene
molecule is added to the chain at the site currently
occupied by the catalyst, (iii) isomerization (chainwalking) in which the catalyst can “walk” to an adjacent
site on the polymer chain, and (iv) chain transfer, which
is thought to be rare. Because the location of the catalyst
controls the position of the next monomer addition,
nonlinear chain propagation (branching) occurs when
monomers add after the catalyst has “walked” to the
middle of a chain strand. The chain-walking process can
be regulated by ethylene pressure, and thus the topology
can be varied from a nearly linear structure with sparse
branching to a densely branched structure by simply
changing the ethylene pressure P.7-9
The plateau modulus, G0N, is an important material
parameter and is one of the characteristic constants for
each type of polymer melt. It reflects the molecular
architecture of the polymer and determines the entanglement molar mass Me. The plateau modulus can
also be correlated with unperturbed chain dimen-
10.1021/ma051408p CCC: $30.25 © 2005 American Chemical Society
Published on Web 11/12/2005
10572
Patil et al.
sions,1,10 such as the radius of gyration Rg. For linear
polyethylene10 (PE), G0N ) 2.60 MPa and Me ) 1040.
For the polymers produced by the catalyst in Figure 1,
the plateau modulus is more pronounced at high ethylene pressure because sparser branching allows the
polymers to interpenetrate and entangle more. If the
connection between molecular architecture and rheology
can be understood, the desirable ability to tailor both
processing and end-use properties in, for example,
polyethylene film becomes a possibility. The objectives
of this research were the synthesis and molecular and
rheological characterization of PEs with similar high
molar masses and narrow molar mass distributions to
determine the influence of branching structure on
rheological properties.
We first discuss the results of a simple computer
simulation of the polymerization (details are in the
Appendix) that provide pictures of the architectures
produced by our polymerization. We then present the
synthesis, molecular characterization, and rheology of
the branched polyethylenes we have made. Connections
are then made between rheology and molecular structure.
Polymerization Simulation
When considering the experimentally determined
properties of the polymers in this study, it is useful to
have in mind the possible molecular structures that
might be formed at different ethylene pressures, which
are consistent with the proposed chain-walking mechanism for the polymerization.4-9 To this end, we have
performed stochastic simulations of the chain growth
based upon this simple mechanism. Our simulation is
described in the Appendix and requires just two parameters, which are the probability of adding a monomer when the catalyst is at a chain end (thus growing
linear chain) or of adding a monomer when the catalyst
is in the middle of a chain section (thus adding a branch
point). If a monomer is not added, then the catalyst
“walks” to an adjacent carbon. As described in the
Appendix, the ratio of these probabilities determines the
branching density and is held fixed to give a branching
density of about 100 branches/1000C. The magnitudes
of these probabilities vary with ethylene pressure, and
this controlling parameter changes the global structure
of the polymers made. Different simulations are performed with different “normalized pressure” P̃ relative
to some (unknown) reference pressure, Pref.
These simulations are very similar in structure to
those described previously by Chen et al.11 and by
Michalak and Ziegler.12 These authors include several
parameters that allow detailed prediction of the exact
proportions of different types of short chain branches,
for comparison with NMR data. This paper is concerned
predominantly with rheology and light scattering, which
are sensitive to the large-scale shape of the polymers.
The simple simulation outlined above allows us to focus
on the few key parameters that affect the large-scale
polymer shape but do not reproduce the NMR data. We
did some test simulations with additional parameters
as used by Chen et al.11 and found that there was no
qualitative difference in the results reported belowsfor
the purposes of clarity, we have omitted these extra
parameters in the subsequent discussion.
To characterize the type of large-scale structures
being formed at different molar masses by our algorithm, we began by calculating the radius of gyration
Macromolecules, Vol. 38, No. 25, 2005
Figure 2. Simulation results for the dependence of radius of
gyration on number of carbons, assuming chain-walking for
normalized ethylene pressure P̃ varying from 10 to 3200.
of the structures, assuming ideal Gaussian chain conditions (i.e., neglecting the swelling effect due to excluded
volume). Although this is not particularly realistic as
regards modeling the solution properties of the molecules, it is a quick way of identifying the molecular
structure that is formed and avoids lengthy Monte Carlo
simulation of the accessible configurations of large
molecules.
Figure 2 is a log-log plot of the radius of gyration vs
number of carbons for molecules made at a range of
values of the pressure parameter, P̃. (These curves
represent an average over 50 different molecules made
at each pressure.) These curves typically show three
distinct regimes. At low molar masses, there is a small
region with slope of (roughly) 1/2, indicative of short,
predominantly linear chains with essentially no branches.
Above 10 carbons (recall, the branching density λ ) 0.1)
the curve adopts a shallower slope, indicating a branched
structure. However, at higher molar masses the slope
increases back to 1/2, indicating that at the largest scales
the structure becomes linear again. We find that, in the
initial linear and branched regimes, the radius of
gyration as a function of molar mass is determined only
by the branching density. As ethylene pressure is varied,
the crossover to the high molar mass region is changed,
and at sufficiently high pressure the branched regime
is absent altogether. We believe that the high molar
mass “linear” regime is present at all pressures, provided one looks to sufficiently high molar masses. These
observations are consistent with, but add detail to,
earlier works11,12 which indicate that the structure
becomes “more linear” and has a larger radius of
gyration as ethylene pressure is increased.
Figures 3 and 4 show, at various molar masses,
molecules growing at P̃ ) 100 and P̃ ) 400, respectively.
These illustrate the last two regimes of growth. (The
eventual linear structure is emphasized by slightly
biasing the step direction of the bonds.) In the second,
branched regime, the catalyst is able to diffuse (via
chain-walking) over the whole molecule as it adds
monomers to the growing structure. The crossover from
the second to the third regime occurs when, as the
molecule gets larger, the catalyst does not have time to
walk over the whole molecule as it adds more monomers
and so gets stuck on one side of the molecule. More
quantitatively, we believe that the crossover occurs at
Macromolecules, Vol. 38, No. 25, 2005
Rheology of Polyethylenes 10573
Figure 3. Simulated polymers produced at normalized ethylene pressure P̃ ) 100 (cf. Figure 2). The polymerization proceeds
from the upper left to the lower right, as polymers have 2000, 4000, 10 000, 20 000, 40 000, 100 000, 200 000, and 400 000 carbons.
degree of polymerization N, where the catalyst walking
time over a molecule of this size equals the time to add
the next N monomers. Increasing ethylene pressure
increases the rate of monomer addition and so pushes
the crossover to lower molar mass (compare Figures 3
and 4). From then onward, the catalyst is rarely able
to retrace its path back along the whole molecule, so
monomers are only ever added to one side of the
molecule, and ultimately a large-scale linear structure
(albeit highly branched at a local level) is formed.
The samples in this study are all made with roughly
the same molecular weight (with roughly 26 000 carbons) but at different ethylene pressures. On the basis
of these simulations, we would anticipate that at low
ethylene pressures the molecules have a highly branched,
globular structure. On increasing the pressure, at fixed
molecular weight, the structure should become globally
“linear”, i.e., the locally branched “sausage-like” structure shown in Figure 3. Further pressure increases lead
the “sausage” to become longer and thinner, giving an
increase in radius of gyration. There is no evidence in
the simulations for an intermediate “hyperbranched”
structure (i.e., long linear-like sections joined by sparse
long-chain branches), as has been suggested previously.7,8
Experimental Section
Polymerization Procedure.4-9 The chain-walking catalyst used in our ethylene polymerization, catalyst 1 (Figure
1), was prepared by following a literature procedure.5,6 Ethylene polymerization was performed in a Parr pressure reactor
equipped with mechanical stirring and temperature control.
A typical polymerization procedure is described as follows: The
Parr reactor was preheated to 100 °C for 2 h under vacuum
and then cooled under N2 to room temperature before use. A
preprepared catalyst 1 solution in chlorobenzene/toluene (1:
3) was transferred into the Parr reactor under N2. After
purging the reactor with N2 and ethylene, the system was filled
with ethylene to the desired pressure, and polymerization was
allowed to continue at room temperature for 48 h. The
polymerization was quenched by addition of excess triethylsilane under N2. The solution was passed through Celite and
neutral alumina gel to remove residual catalyst and then
concentrated in vacuo. The polymer was finally precipitated
in methanol or acetone and dried under vacuum.
SEC-MALLS Characterization of Copolymers.7,8 All the
polymers were characterized by size-exclusion chromatography
(SEC) coupled to a multiangle laser light scattering detector
(MALLS) for obtaining both the weight-average molar mass
(Mw) and the z-average radius of gyration (Rg). Measurements
were made on dilute fractions eluting from a SEC consisting
of a HP Aglient 1100 solvent delivery system/auto injector with
an online solvent degasser, temperature-controlled column
compartment, and an Aglient 1100 differential refractometer.
A Dawn DSP 18-angle light scattering detector (Wyatt Technology, Santa Barbara, CA) was coupled to the SEC to measure
both Mw and Rg for each fraction of the polymer eluted from
the SEC column. A 30 cm column was used (Polymer Laboratories PLgel Mixed C, 5 µm particle size) to separate polymer
samples. The mobile phase was uninihibited tetrahydrofuran,
for which polyethylenes have dn/dc ) 0.078 mL/g and the flow
rate was 0.5 mL/min. Both the column and the differential
refractometer were held at 35 °C. 60 µL of a 2 mg/mL solution
was injected into the column. Acetone (0.2%) was used as an
internal flow marker. Results for Mn, Mw, Mz, Mw/Mn, and Rg
are listed in Table 1.
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Patil et al.
Macromolecules, Vol. 38, No. 25, 2005
Figure 4. Simulated polymers produced at normalized ethylene pressure P̃ ) 400 (cf. Figure 2). The polymerization proceeds
from the upper left to the lower right, as polymers have 200, 400, 1000, 2000, 4000, 10 000, 20 000, and 40 000 carbons.
on the following equation:
Table 1. Characterization Results
pressure (atm)
Mn
Mw
Mz
Mw/Mn
Rg (nm)
branches/1000 CH2
Tg (°C)
η0 (Pa s) at 30 °C
G0N (MPa) at 0 °C
0.1
235K
373K
553K
1.58
13.0
97
-75
114
0.3
330K
441K
603K
1.33
16.4
97
-68
125
1.0
318K
481K
705K
1.51
23.5
96
-67
67000
10.2
378K
486K
643K
1.28
25.9
95
-65
5.7 × 106
0.10
34.0
447K
518K
592K
1.16
31.2
92
-56
1.1 × 108
0.74
DSC Determination of Glass Transition. Glass transition temperature (Tg) of the PE samples was determined using
a Perkin-Elmer differential scanning calorimeter (Pyris 6). The
PE samples were prepared for DSC measurements by drying
under vacuum at 60 °C for 3 days. Liquid nitrogen was used
as coolant, and nitrogen was used as purging gas for the
experiments. The scanning temperature range was -90 to 200
°C at a scanning rate of 10 °C/min. Results for Tg are listed in
Table 1.
NMR Determination of Branching Density. NMR spectra were recorded on Bruker DRX400, Bruker GN500, or
Bruker Omega500 MHz FT-NMR instruments. The PE samples
were prepared for NMR measurements by drying under
vacuum at 60 °C for 3 days and then dissolved in deuterated
chloroform (CDCl3) at a concentration of 10 mg/mL.
Branching density of the PE samples was determined from
their NMR spectra using literature methods.13 Since one
branching point on the polymer chain will inevitably result in
one more end methyl group -CH3 (no loop structure), the
branching density per 1000 carbons can be determined by
measuring the number of methyl groups per 1000 carbons from
1H NMR spectra. Protons from CH groups can be precisely
3
determined because they are clearly separated from protons
from CH and CH2 groups. The branching density Nbr (branches
per 1000 carbon atoms) is reported in Table 1, calculated based
500
Nbr )
[
no. of CH3 protons
-2
3
total no. of protons
]
(1)
Rheology. The PE samples were prepared for rheology
measurements by drying under vacuum at 60 °C for several
days, followed by compression-molding an 8 mm diameter, ∼1
mm thick disk at 80 °C, for solidlike samples synthesized at
high P. Samples synthesized at low P were sufficiently fluid
at room temperature to be poured directly onto the bottom
plate of the rheometer after drying. Oscillatory shear tests
were performed using a Rheometrics strain-controlled ARES
rheometer. The angular frequency ω was varied from 102 to
10-4 rad s-1, and the temperature ranged from -60 to 70 °C.
The temperature was decreased in 30 K increments at high
temperatures and at 5 K intervals at temperatures approaching Tg. All PE samples were studied under a nitrogen
atmosphere in order to minimize thermooxidative degradation.
The isothermal rheology data at different temperatures were
shifted to obtain master curves at the reference temperature
T0 ) 30 °C using time-temperature superposition.1
Results
Linear and branched structures have different swollen fractal dimensions1 and consequently exhibit different exponents on plots of log Rg vs log Mw. Linear
polymers have exponent ν ≡ d(log Rg)/d(log Mw) ≈ 3/5,
while branched polymers have exponent ν ≈ 1/2 in good
solvent. While NMR shows that the branching density
is similar in all samples, the increase in Rg with
ethylene pressure indicates a change in the arrangement of these branches, toward a more open structure
with increasing ethylene pressure. This is seen in Figure
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Rheology of Polyethylenes 10575
Figure 5. Correlation of radius of gyration with weightaverage molar mass (both obtained from SEC-MALLS) for
polyethylene samples made at 0.1 atm (open circles; ν ) 0.53),
0.3 atm (filled circles, ν ) 0.48), 1.0 atm (open squares; ν )
0.46), and 34 atm (filled triangles; ν ) 0.41, all exponents with
uncertainties of (0.02).
Figure 6. Master curves of the frequency dependence of
storage modulus G′(ω) (filled symbols) and loss modulus G′′(ω) (open symbols) of unentangled samples synthesized at
ethylene pressures of 0.1 atm (squares) and 0.3 atm (circles)
at the reference temperature T0 ) 30 °C. The data for the
sample prepared at 0.3 atm have been shifted vertically by a
factor of 10 for clarity. The line has the slope 3/5 expected by
the Rouse model for a polymer with fractal dimension D ) 3.
5, where the magnitude of Rg increases with ethylene
pressure. We note that the exponent ν decreases with
ethylene pressure (see the Figure 5 caption for the
exponent values). The exponent ν ranges from 0.41 to
0.53, similar to experimental values for randomly
branched polymers in good solvent.
Figure 6 shows the frequency dependence of storage
and loss moduli, G′(ω) and G′′ (ω), at a reference
temperature of 30 °C, for samples made at low ethylene
pressure (P ) 0.1 and 0.3 atm). Densely branched
polymers are formed at these low pressures, and the
melt rheology indicates no entanglement even though
Mw ) 3.7 × 105. This suggests that the polymers made
at low ethylene pressure do not overlap enough to
entangle, as there is no hint of any rubbery plateau in
Figure 6. The branched polymer Rouse model1,14 predicts a power law in G′(ω) and G′′ (ω) with exponent
determined by the fractal dimension of the polymers (see
eq 8.148 of ref 1).
G′ ∼ G′′ ∼ ω3/(D+2) ∼ ω3/5
(2)
The final result was obtained using the expected fractal
dimension of 3, owing to the crowded nature of these
highly branched molecules.15 The line in Figure 6 has
slope 3/5, demonstrating that the branched polymer
Rouse model describes the linear viscoelasticity of these
Figure 7. Master curves of the frequency dependence of
storage modulus G′(ω) (filled symbols) and loss modulus
G′′(ω) (open symbols) of entangled samples synthesized at
ethylene pressures of 1 atm (squares), 10.2 atm (circles), and
34 atm (triangles) at the reference temperature T0 ) 30 °C.
For clarity, the data for the sample prepared at 10.2 atm have
been shifted vertically by a factor of 10, and the data for the
sample prepared at 34 atm have been shifted vertically by a
factor of 300.
unentangled highly branched polymers. The sample
prepared at 0.1 atm has G′ ∼ G′′ ∼ ω0.62(0.02 for the
highest 2 decades of frequency, and the sample prepared
at 0.3 atm has G′ ∼ G′′ ∼ ω0.59(0.02 for the highest 2.5
decades in frequency in Figure 6.
Randomly branched condensation polymers that are
not entangled exhibit a power law viscoelastic response14 qualitatively similar to that observed in Figure
6 at high frequencies (103 rad/s < ω < 105 rad/s). The
frequency exponent of this power law for randomly
branched polymers is also understood using eq 2, but
with smaller fractal dimension of 2.53, giving a larger
frequency exponent of 0.66. Randomly branched polymers also have a much broader (power law) molar mass
distribution,1,14,16 but since the unentangled randomly
branched polymers have their dynamics described by
the Rouse model1,14 (with no interactions between
chains), the frequency exponent is insensitive to the
distribution and only depends on the fractal character
of the molecules. Consequently, this simple analysis also
holds for our highly branched polymers produced at low
pressures, with quite narrow molar mass distributions.
The samples made at low ethylene pressure are more
like dendrimers, which are also known to not interpenetrate1 and not entangle.17,18
In the melts of densely branched polymers, entanglements do not occur because of the limited possibility of
interpenetration between neighboring molecules. The
relatively high branching density leads to a compact
structure of deformable individual macromolecules.
Linear viscoelastic response of randomly branched
condensation polymers, with average chain lengths
between branch points shorter than 2Me, are known to
be described by the branched polymer Rouse model.14
Systems with longer linear chain sections between
branch points systematically deviate from the Rouse
predictions,16 presumably indicating the importance of
interchain entanglements.
Polymers produced at higher ethylene pressures have
only slightly higher molar masses but show entanglement effects in their rheology, presumably because they
have a more open structure that allows for interpenetration and interchain entanglement. Figure 7 shows
a clear rubbery plateau region in G′(ω) (in the frequency
range 1 rad/s < ω < 106 rad/s for the sample made at P
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Patil et al.
Macromolecules, Vol. 38, No. 25, 2005
) 34 atm) with weak frequency dependence of G′(ω) and
G′(ω) . G′′(ω), indicating that the polymers produced
at high ethylene pressure are entangled. Two easily
distinguishable relaxation processes are evident in
Figure 7: (1) the transition zone1 involving segmental
relaxation at high frequencies, which separates the
glassy modulus of the polymer from the rubbery plateau,
and (2) chain relaxation at lower frequencies. Some
variations in the transition zone are caused by the
observed differences in glass transition temperatures
(see Table 1).
The appearance of a rubbery plateau in Figure 7
indicates an elastic character of the polymer melt,
caused by chain entanglement. However, the plateau
modulus seen is much lower than that of linear PE. For
example, Figure 7 shows that the sample made at an
ethylene pressure P ) 34 atm is strongly entangled,
with a plateau modulus of order 1 MPa, somewhat lower
than conventional polyethylene (G0N ) 2.6 MPa) because of the high level of short chain branching (roughly
every 10th carbon atom). The molar mass of an entanglement strand,1 Me, can be calculated from the
plateau modulus
Me )
FRT
G0N
Figure 8. Frequency dependence of the complex viscosity at
30 °C, for samples synthesized at ethylene pressures of P )
0.1, 0.3, 1.0, and 10.2 atm.
(3)
where R is the universal gas constant (R ) 8.314 J mol-1
K-1), F is the density of the polymer at temperature T,
and G0N is the plateau modulus. Since the plateau
modulus is lower, the values of Me obtained are larger
than conventional linear PE. Only the samples made
at the highest ethylene pressures had rubbery plateaus
that were sufficiently well-defined to exhibit a minimum
in G′′, and the plateau modulus was estimated as the
value of G′ at the frequency where G′′ shows a local
minimum (see Table 1). For the sample made at an
ethylene pressure of 34 atm, G0N ) 0.74 MPa at 0 °C
and Me ) 2500, while the sample made at an ethylene
pressure of 10.2 atm has G0N ) 0.10 MPa at 0 °C and
Me ) 18 000.
Time-temperature superposition works very nicely
for the samples prepared at low ethylene pressures
(Figure 6) but works more poorly as ethylene pressure
is increased (Figure 7). Failure of superposition was
observed most clearly for G′′(ω) in the plateau region
of the samples made at high ethylene pressures. Since
NMR suggests similar levels of short-chain branching
(92-97 branches per 1000 carbons) in all the samples,
large-scale architectural differences in structure must
account for differences in entanglement and timetemperature superposition in these branched polyethylenes. As ethylene pressure is increased, subtle changes
in chain overlap with temperature apparently affect the
extent of entanglement slightly.
The frequency dependence of |η*(ω)| (magnitude of
complex viscosity) is plotted in Figure 8 for four representative samples at a reference temperature of 30 °C.
The low shear rate value of the complex viscosity is the
zero-shear-rate viscosity η0. Table 1 lists the values of
η0 at 30 °C for the various samples studied: The zeroshear-rate viscosity increases by 6 orders of magnitude
when the ethylene pressure is increased, even though
the molar mass is nearly constant, as shown in Figure
9. We can understand the increase in viscosity for the
entangled polymers with a simple scaling argument.
The simulations suggest that the overall architecture
Figure 9. Dependence of zero-shear-rate viscosity at 30 °C
on the ethylene pressure used in synthesis.
of these polymers is a linear backbone with considerable
amounts of short-chain branched fuzz. Since the branching is roughly every 10 carbons, and polyethylene has
75 carbons per entanglement strand, the branching is
too dense to allow entanglements to form in the fuzz.
We therefore envision only the linear backbones being
able to entangle, and the fuzz simply acts to dilute the
backbones, giving the backbones an effective volume
fraction φ. This volume fraction can be directly determined from the experimental plateau modulus G0N
φ2 )
G0N Me,0
)
G0
Me
(4)
where G0 ) 2.60 MPa is the plateau modulus of linear
polyethylene,10 Me is the experimental value of the
entanglement molar mass, and Me,0 ) 1040 is the molar
mass of an entanglement strand for linear polyethylene.10 On the basis of the measured plateau moduli and
eq 4, we conclude that φ ) 0.64 for the sample prepared
at 34 atm and φ ) 0.24 for the sample prepared at 10.2
atm. This backbone volume fraction also affects the endto-end distance of the backbone R and assuming Gaussian statistics, R2 ∼ φM. The radius of gyration data in
Figure 5 suggest the sample prepared at 34 atm has
roughly twice the size of the sample prepared at 1 atm,
suggesting a crude estimate for the backbone volume
fraction of the 1 atm sample, φ ≈ 0.64/22 ) 0.16. This
estimate is consistent with the expectation of eq 4, based
on an apparent plateau modulus of G0N ) 70 000 Pa for
the 1 atm sample in Figure 7. The tube diameter a is
determined from random walk statistics along the
backbone, with effective entanglement molecular weight
along the backbone Me,bb smaller than Me by a factor
Macromolecules, Vol. 38, No. 25, 2005
Rheology of Polyethylenes 10577
that is the backbone volume fraction φ.
a2 ∼ Me,bb ) φMe )
Me,0
φ
(5)
Reptation of the linear backbones (covered with fuzz)
occurs by curvilinear diffusion along the tube of contour
length L ) Ma/Me with curvilinear diffusion coefficient
Dc that depends on M but not φ. The tube disengagement time1
τ)
M2a2
L2
∼
∼ φ3
Dc M 2D
e
(6)
c
is the longest relaxation time in the polymer melt. The
product of the relaxation time and the plateau modulus
determines the melt viscosity
η ) G0Nτ ∼ φ5
(7)
which scales as the fifth power of the volume fraction.
This explains the factor of 1000 change in the viscosity
of the entangled samples in Figure 9 when the ethylene
pressure is changed from 1 to 34 atm, as (0.64/0.16)5 )
1000.
Comparison of Simulations and Experiments. At
a qualitative level, there are a number of aspects of the
experimental results that seem very consistent with the
predicted molecular structures emerging from the simulations. Simulations suggest that the branching density
should remain constant with ethylene pressure, which
is observed. The predicted “globular” structures at low
ethylene pressure would be anticipated to exhibit unentangled polymer rheology, as is the case. At higher
pressures, the simulations predict a linear structure,
which is highly branched at a local level but which gets
longer and thinner with increasing pressure. One would
expect, as the molecules become globally more “linear”,
that they would begin to become entangled in the melt
state and that the longer, thinner structures would have
a higher plateau modulus. This is very consistent with
the increasing plateau modulus and decreasing entanglement molecular weight observed with increasing
pressure in the experiments. Finally, the simulations
indicate an increasing radius of gyration with increasing
ethylene pressure, and this is observed in light scattering results.
Nevertheless, several puzzles remain in the experimental data that we have not been able to account for.
The light scattering results are taken in good solvent
conditions. The observed slope in the plot (Figure 5) of
log Rg vs log Mw is about 0.5 at the lowest pressure,
decreasing to about 0.41 at the highest pressure. In good
solvent conditions, a slope of 0.5 is indicative of a
branched, not a linear, structure. In the simulations,
the slope is always predicted to increase with increasing
pressure, toward a value commensurate with a linear
topology (such a slope would be about 3/5 in good
solvent). Thus, there is no experimental evidence of
large-scale linear structures in the light scattering data.
There are also unexplained features in the rheological
measurements. The polymer made at 10.2 atm has a
plateau modulus roughly a factor of 7 lower than the
34 atm polymer. Despite this, its terminal time appears
to be roughly a factor of 10 larger than the 34 atm
polymer. This observation is very difficult to explain
using conventional tube models for entangled linear
polymers; one would expect that the polymer with the
lower entanglement molecular weight would lie within
a longer, thinner tube and consequently have a longer
terminal time. It might be that there are some “long
chain branches” in the 10.2 atm polymer, which would
explain the larger terminal time. Some qualitative
evidence for this is also present in the broad width of
the “peak” in the loss modulus. However, long chain
branches are not really consistent with the polymerization simulations based on the chain-walking mechanism.
A final puzzle lies in the rate at which the polymer
properties change with increasing ethylene pressure.
According to the simulations (see Figure 2), for polymers
with between 104 and 105 carbons, most of the structural
changes occur between P̃ ) 40 and P̃ ) 1600, a factor
of (roughly) 40 in pressure. Outside this range, the
simulations predict very small changes occur in the
global structure for the molar masses studied in the
experiments. In the experiments, however, the structure
appears to vary continuously between polymers made
at pressures of 0.1 atm and those made at 34 atm, a
factor of 340 in pressure. This discrepancy might be
explained by non-first-order reaction kinetics with
respect to ethylene pressure. Taken together, there are
enough discrepancies to prevent the molecular structures emerging from the stochastic simulation being
used to predict the polymer properties. This leads us to
question whether there is important science missing
from the simulations in their present form.
One important ingredient that is lacking is the effect
of excluded volume. This will affect the radius of
gyration in the solution state. (Chen et al.11 addressed
this issue for relatively smaller molecules with, typically, 5000 carbons.) Excluded volume could also affect
the polymerization process itself. In highly branched
structures, excluded volume might prevent the bulky
catalyst from walking to all possible sites. By inhibiting
chain-walking, excluded volume would slow down catalyst diffusion along the polymer contour. This creates
more linear structures, since the catalyst does not
diffuse as far before the next reaction. These suggestions
are borne out by some simple test simulations we have
performed, in which polymers are laid down on a cubic
lattice and the catalyst is prevented or discouraged from
walking to “crowded” sites on the lattice. However, a
fully realistic simulation, in which both the catalyst and
growing polymer are given sensible dynamics, would be
quite costly and is beyond the present investigation.
However, including excluded-volume effects does not
even qualitatively account for two of the above discrepancies: the slope of log Rg vs log Mw (which indicates a
globally branched, rather than a linear, structure) and
the relatively large terminal time of the 10.2 atm
polymer (which is also suggestive of some large-scale
branching). The chain-walking mechanism alone cannot
predict structures that are branched on a large scale; it
leads only to linear structures, as shown in Figures 2,
3, and 4. The experimental evidence we have at present
is therefore suggestive of the need for some mechanism
in addition to chain-walking during the polymerization
process.
Conclusion
The molecular structure and rheology of branched
polyethylenes synthesized with a palladium-bisimine
catalyst change systematically with ethylene pressure.
10578
Patil et al.
Macromolecules, Vol. 38, No. 25, 2005
Low ethylene pressures allow the catalyst to walk
extensively along the molecule between additions of
ethylene and produce densely branched polymers that
do not entangle. The linear viscoelastic response of these
unentangled densely branched polymers is well described by the Rouse model. As ethylene pressure is
increased, ethylene addition occurs faster, not allowing
the catalyst time to walk as far, leading to more open
structures that form entanglements in the melt. Our
current understanding of the chain-walking mechanism
suggests that the molecules produced at high ethylene
pressures should be densely branched on their smallest
scales, and be linear on their largest scales, with no
long-chain branching. However, both the swollen fractal
dimension of the molecules and their rheology suggest
that long-chain branching effects are observed. Consequently, mechanisms other than chain-walking should
be considered in the polymerization.
Acknowledgment. This material is based upon
work supported by the U.S. Army Research Laboratory
and U.S. Army Research Office under Grant DAAD1902-1-0275 Macromolecular Architecture for Performance
(MAP) MURI. Z.G. thanks the National Science Foundation (DMR-0135233) for financial support of this
work.
Appendix: Computer Simulation
A molecule is grown as follows: when the catalyst is
attached to a given carbon, I, there is a site-dependent
probability pI that the next significant action of the
catalyst is to add a monomer to that site. The alternative, with probability 1 - pI, is to “walk” to an adjacent
carbon in the same molecule, chosen randomly from its
neighbors. We distinguish between three different sites
on the molecule. If the catalyst is at a chain end (i.e., a
carbon with one neighbor), then the addition of a
monomer (with probability pend) continues the growth
of a linear section of chain. If the catalyst is in the
middle of a linear section of chain (i.e., a carbon with
two neighbors), then the addition of a monomer (with
probability pmid) adds a side branch. If the catalyst is
on a branch point (i.e., a carbon with three neighbors),
then we do not allow monomer additionsthe catalyst
jumps to one of its neighbors. This algorithm allows the
growth of a molecule up to a specified number of carbons
and requires just two parameters, pend and pmid.
The probabilities pend and pmid vary with ethylene
pressure; we can predict the form of this variation using
a simple kinetic model. The catalyst is thought to exist
in two states: a trapped state (T) in which it traps an
ethylene monomer, with the possibility of adding it to
the growing chain, and an untrapped state (U) in which
it might walk to an adjacent site.7-9 Beginning in the
untrapped state U, a catalyst might walk to an adjacent
site U′ or trap a monomer. These can be represented
schematically as
U f U′
kW,I
UfT
kT,IP
where kW,I and kT,I are the rate constants for the two
processes for site I, and the rate of the second process
is proportional to ethylene pressure P. Hence, in the U
state, the probabilities of “walking” or “trapping” are
pW,I )
kW,I
kW,I + kT,IP
pT,I )
kT,IP
kW,I + kT,IP
In the trapped state, T, the monomer might be added
to the chain or dissociate from the catalyst, represented
as
T f ∼U′′
TfU
kA,I
kD,I
where the notation ∼U′′ is used to indicate a new
unbound state in which a monomer has just been added.
In the T state, the probabilities of “addition” or “dissociation” are
pA,I )
kA,I
kA,I + kD,I
pD,I )
kD,I
kA,I + kD,I
Starting from a given U state, there are hence three
possibilities, all of which return the catalyst to a U
state: walking, trapping followed by monomer addition,
and trapping followed by dissociation. The latter returns
the catalyst to the original U state and so does not
constitute a significant “action”. Hence, the probability
pI of monomer addition being the next significant action,
starting from the U state, is
pI )
)
pT,IpA,I
1 - pT,IpD,I
rIP̃
1 + rIP̃
(8)
where
rI )
pA,IkT,IPref
kW,I
and
P̃ )
P
Pref
Here, Pref is a reference pressure. Hence, if we specify
rend and rmid at some low value of the reference pressure,
then eq 1 indicates how pend and pmid will vary with
normalized pressure, P̃.
It is clear, from the above equations, that increasing
ethylene pressure increases the rate of monomer addition relative to the rate of chain-walking. At sufficiently
low pressures chain-walking will be the dominant
mechanism, so that the catalyst will sample many “mid”
and “end” sites during the addition of monomers. In this
limit, we obtain a simple formula relating the branching
density, λ (defined here as the number of branches per
carbon), to the simulation parameters pend and pmid. In
a highly branched structure, a fraction λ of carbons are
branch points, another fraction λ of carbons are “end”
Macromolecules, Vol. 38, No. 25, 2005
Rheology of Polyethylenes 10579
sites, and the remaining fraction (1 - 2λ) are “mid” sites.
Assuming the catalyst samples an average over these
sites when adding monomers, the rate of increase of
number B of branch points and N of carbons with
simulation steps n is
approaches 1. In our simulations, we attached the
catalyst to the penultimate carbon, so as to preserve a
constant value of λ over as large a range of pressure as
possible.
dB
) pmid(1 - 2λ)
dn
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dN
) 2[pmid(1 - 2λ) + pendλ]
dn
At steady state, we must have λ ) dB/dN, which gives
a quadratic formula for λ that we solve to obtain
λ)
1
x
pend
+2
pmid
(9)
In the simulations reported in this paper, we set rend )
0.000 62 and rmid ) 0.000 01 and allow P̃ to vary. This
gives branching densities of order 0.1 (100 branches per
1000 carbons) in all the simulations reported herein,
thereby matching the experiments.
Equation 9 potentially breaks down at the higher
pressures when first pend and then pmid approach 1, and
monomer addition dominates over chain-walking. One
detail that is important here is whether, after the
addition of a monomer, the catalyst is attached to the
end site on that monomer or to the penultimate carbon.
If the catalyst is attached to the end carbon, then eq 9
breaks down when pend approaches 1. If, on the other
hand, the catalyst is attached to the penultimate carbon,
then eq 9 breaks down at a higher pressure, when pmid
References and Notes
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